How to fix strange behavior of vertexes? (picture)
Your vertices are weighted on bad bones.

The selected vertex should have 1 for Bone1.035 and nothing else. This can be seen in Edit mode in the N panel. From there you can change the weight values (in this case, 1 for Bone1.035 and remove with X bone3.044).
The other vertex needs to have only bone3.044.
Once this is done, place your vertices where they should be and they'll do what you expect.

You're life would be easier with better bones names (Something like Arm1.bone1, arm1.bone2, Arm2.bone1 ...)